International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A1, ch. 1.2, pp. 11-12
Section 1.2.4. Subgroups^{a}Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany |
There may be sets of elements that do not constitute the full group but nevertheless fulfil the group postulates for themselves.
Definition 1.2.4.1.1. A subset of elements of a group is called a subgroup of if it fulfils the group postulates with respect to the law of composition of .
Remarks:
Definition 1.2.4.1.2. A subgroup is a maximal subgroup if no group exists for which holds. If is a maximal subgroup of , then is a minimal supergroup of .
This definition is very important for the tables of this volume, as only maximal subgroups of space groups are listed. If all maximal subgroups are known for any given space group, then any general subgroup can be obtained by a (finite) chain of maximal subgroups between and , see Section 1.2.6.2. Moreover, the relations between a space group and its maximal subgroups are particularly transparent, cf. Lemma 1.2.8.1.3.
Let be a subgroup of of order . Because is a proper subgroup of there must be elements that are not elements of . Let be one of them. Then the set of elements ^{3} is a subset of elements of with the property that all its elements are different and that the sets and have no element in common. Thus, the set also contains elements of . If there is another element which belongs neither to nor to , one can form another set . All elements of are different and none occurs already in or in . This procedure can be continued until each element belongs to one of these sets. In this way the group can be partitioned, such that each element belongs to exactly one of these sets.
Definition 1.2.4.2.1. The partition just described is called a decomposition ( : ) into left cosets of the group relative to the group . The sets are called left cosets, because the elements are multiplied with the new elements from the left-hand side. The procedure is called a decomposition into right cosets if the elements are multiplied with the new elements from the right-hand side. The elements or are called the coset representatives. The number of cosets is called the index of in .
Remarks:
From its definition and from the properties of the coset decomposition mentioned above, one immediately obtains the fundamental theorem of Lagrange (for another formulation, see Chapter 1.5 ):
Lemma 1.2.4.2.2. Lagrange's theorem: Let be a group of finite order and a subgroup of of order . Then is a divisor of and the equation holds where is the index of in .
A special situation exists when the left and right coset decompositions of relative to result in the partition of into the same cosets: Subgroups that fulfil equation (1.2.4.1) are called `normal subgroups' according to the following definition:
Definition 1.2.4.2.3. A subgroup is called a normal subgroup or invariant subgroup of , , if equation (1.2.4.1) is fulfilled.
The relation always holds for , i.e. subgroups of index 2 are always normal subgroups. The subgroup contains half of the elements of , whereas the other half of the elements forms `the other' coset. This coset must then be the right as well as the left coset.
In a coset decomposition, the set of all elements of the group is partitioned into cosets which form classes in the mathematical sense of the word, i.e. each element of belongs to exactly one coset.
Another equally important partition of the group into classes of elements arises from the following definition:
Remarks:
Not only the individual elements of a group but also the subgroups of can be classified in conjugacy classes.
Definition 1.2.4.3.2. Two subgroups are called conjugate if there is an element such that holds. This relation is often written .
Remarks:
Equation (1.2.4.1) can be written Using conjugation, Definition 1.2.4.2.3 can be formulated as
For the following definition, the `product of sets of group elements' will be used:
Definition 1.2.4.4.1. Let be a group and , be two arbitrary sets of its elements which are not necessarily groups themselves. Then the product of and is the set of all products .^{4}
The coset decomposition of a group relative to a normal subgroup has a property which makes it particularly useful for displaying the structure of a group.
Consider the coset decomposition with the cosets and of a group relative to its subgroup . In general the product of two cosets, cf. Definition 1.2.4.4.1, will not be a coset again. However, if and only if is a normal subgroup of , the product of two cosets is always another coset. This means that for the set of all cosets of a normal subgroup there exists a law of composition for which the closure is fulfilled. One can show that the other group postulates are also fulfilled for the cosets and their multiplication if holds: there is a neutral element (which is ), for each coset the coset forms the inverse element and for the coset multiplication the associative law holds.
Definition 1.2.4.4.2. Let . The cosets of the decomposition of the group relative to the normal subgroup form a group with respect to the composition law of coset multiplication. This group is called the factor group . Its order is , i.e. the index of in .
A factor group is not necessarily isomorphic to a subgroup .
Factor groups are indispensable for an understanding of the homomorphism of one group onto the other. The relations between a group and its homomorphic image are very strong and are expressed by the following lemma:
Lemma 1.2.4.4.3. Let be a homomorphic image of the group . Then the set of all elements of that are mapped onto the unit element forms a normal subgroup of . The group is isomorphic to the factor group and the cosets of in are mapped onto the elements of . The normal subgroup is called the kernel of the mapping; it forms the unit element of the factor group . A homomorphic image of exists for any normal subgroup of .
The most important homomorphism in crystallography is the relation between a space group and its homomorphic image, the point group , where the kernel is the subgroup () of all translations of , cf. Section 1.2.5.4.
The concept of the normalizer of a group in a group is very useful for the considerations of the following sections. The size of the conjugacy class of in is determined by this normalizer.
Let and . Then holds because is a group. If , then for any . If is not a normal subgroup of , there may nevertheless be elements for which holds. We consider the set of all elements that have this property.
Definition 1.2.4.5.1. The set of all elements that map the subgroup onto itself by conjugation, , forms a group , called the normalizer of in , where .
Remarks: